# Osculating curve

(Redirected from Osculation)
"Osculation" redirects here. For other meanings, see Kiss.
A curve C containing a point P where the radius of curvature equals r, together with the tangent line and the osculating circle touching C at P

In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if F is a family of smooth curves, C is a smooth curve (not in general belonging to F), and p is a point on C, then an osculating curve from F at p is a curve from F that passes through p and has as many of its derivatives at p equal to the derivatives of C as possible.[1][2]

The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency.[3]

## Examples

Examples of osculating curves of different orders include:

• The tangent line to a curve C at a point p, the osculating curve from the family of straight lines. The tangent line shares its first derivative (slope) with C and therefore has first-order contact with C.[1][2][4]
• The osculating circle to C at p, the osculating curve from the family of circles. The osculating circle shares both its first and second derivatives (equivalently, its slope and curvature) with C.[1][2][4]
• The osculating parabola to C at p, the osculating curve from the family of parabolas, has third order contact with C.[2][4]
• The osculating conic to C at p, the osculating curve from the family of conic sections, has fourth order contact with C.[2][4]

## Generalizations

The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a space curve is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.[5]